How to check whether feasible solutions for linear programs do exist?

Question

For a linear program, how to decide whether there exists a feasible solution without solving it?

For $Ax \le b$, is there any sufficient and/or necessary condition represented by matrix $A$ and vector $b$ to decide the existence of a feasible solution without solving the problem?

please state the complete feasible set. If it is ${x|Ax\le B}$ and $B\ge 0$, then $x=0$ is feasible. — user251257, Jul 14 '15 at 10:31
This is just a special case, I need a general answer. Thank you all the same. — Frank, Jul 14 '15 at 11:18
what do you mean by constructive method? I want a method which can easily be implemented on computer, I don't really care about the theory behind. — Frank, Jul 14 '15 at 11:46
That's what I meant. then, in general, you need to implement the simplex method or something similar. — user251257, Jul 14 '15 at 13:46
There is no method that is cheaper than solving the original problem. Look at "two-phase" methods for linear programming; you are looking for phase I. But in fact, you don't need to use two-phase methods; there are methods that determine feasibility and optimality in a single phase, and those are just as cheap (theoretically). — Michael Grant, Jul 15 '15 at 12:31
possible duplicate of How to find out whether linear programming problem is infeasible using simplex algorithm — Michael Grant, Jul 15 '15 at 12:34

A.Γ. · Answer 1 · 2015-07-14T11:44:50.167

4

You can use the following result:

Theorem (Farkas lemma, 1894): Let $A\in\mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$. Then $Ax\le b$ has a solution if and only if $A^Ty=0$, $y\ge 0$, $b^Ty<0$ has no solution.

Corollary: In other words, $A^Ty=0$, $y\ge 0$ $\Rightarrow$ $b^Ty\ge 0$.

edited Jul 14 '15 at 11:44

answered Jul 14 '15 at 11:17

A.Γ.

30,381

thanks, I will have a try. – Frank Jul 14 '15 at 11:26

How to check whether feasible solutions for linear programs do exist?

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